606 
MR. LUBBOCK’S RESEARCHES 
Thus 
J =1 + 3e4 (l + ^)~ 2e ( ] + 4 - e2 ) cos *- 
[0] [2] 
— e 4 cos 2 x + — cos 3 x + A- cos 4 x 
o O o 
[ 8 ] 
[ 20 ] 
[35] 
{»(*—»/) + 3 »} 2 { (1 + 3 « 2 ) r ga - 1 r 10 - j - * e2 + 2 ( ia + 3 a fi 2a + = 0 
L JL 2 1 o J *(«“ # /)+3n da 
If 7? be considered as a function of r', X' and we have 
djR _ rdR cl r' cl fl d^ dR ds 
d y dr r d y dx'dy cl s d y 
~r^= y ~ -J (* — 4e ‘) c ° s 2y + ye cos (a; — 2y) -yecos (x + 2y) 
[62] 
[65] 
[ 66 ] 
— -4 7 e ~ cos (2 x — 2 y) + —ye 2 cos (2 x + 2y) 
b 8 
[77] 
[78] 
djC 
cl y 
_ y ( 1 _ 4 e 2 ) sin 2 ?/ + y e sin (a; — 2 y) — 3 y e sin (a: + 2 y) 
[62] 
[65] 
[ 66 ] 
13 
— y e 2 sin (2a; + 2y) 
[78] 
d 5 • • g- Q 
— = (1 — e 2 ) sin y + esm (x — y) + e sin (x + y) + —sin (2a? — ?/)+ — e 2 sin (2 x + y) 
(1 y o o 
[146] [149] [150] [161] 
If R be considered as a function of r, X' and s , 
[162] 
d R d R dx' d R ds 
cl y d X' d y dsdy 
^ = — i 7f as before, and the expression for ^ (in this case) is given for the 
Lunar Theory, Phil. Trans. 1832, p. 6. 
The multiplications required may be effected by means of the following 
Table. In the terms multiplied by the coefficient of H is to be taken with 
a positive sign when its index is found in the upper line, and with a negative in 
the contrary case. In the terms multiplied by the coefficient of ^ is 
to be taken with a negative sign when the index is found in the upper line, and 
with a positive when in the lower. 
